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Part 1

Suppose that Bank A pays 

2.602.60​% 

interest compounded quarterly on 

aa 

44​-year 

​CD, while Bank B pays 

2.592.59​% 

compounded daily. 

**a.** What are the effective rates for the two​ CDs? Use a​ 365-day year.

**b.** Suppose 

​$50005000 

was invested in each of these accounts. Find the compound amount after 

fourfour 

yearsyears 

for each account.

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Part 1

**a.** The effective rate for Bank A is 

enter your response here​%.

​(Do not round until the final answer. Then round to three decimal places as​ needed.)

To solve this problem, we need to calculate the effective annual rate (EAR) for both banks and then compute the compound amount after four years for an investment of $5000 in each bank.

### Part A: Effective Annual Rate (EAR)

1. **Bank A: Quarterly Compounding**
   - Nominal interest rate: $ r_A = 2.60\% $
   - Compounding frequency: Quarterly (4 times per year)

   The formula for the effective annual rate (EAR) with compounding is:
   \[
   EAR = \left( 1 + \frac{r}{n} \right)^n - 1
   \]
   Where:
   - $ r $ is the nominal interest rate (as a decimal),
   - $ n $ is the number of compounding periods per year.

   For Bank A:
   \[
   EAR_A = \left( 1 + \frac{0.0260}{4} \right)^4 - 1
   \]

2. **Bank B: Daily Compounding**
   - Nominal interest rate: $ r_B = 2.59\% $
   - Compounding frequency: Daily (365 times per year)

   Similarly, for Bank B:
   \[
   EAR_B = \left( 1 + \frac{0.0259}{365} \right)^{365} - 1
   \]

We will now calculate both EARs using these formulas.

### Part B: Compound Amount After Four Years

For the compound amount, we use the formula:
\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]
Where:
- $ A $ is the amount after $ t $ years,
- $ P $ is the principal amount ($5000),
- $ r $ is the nominal interest rate (as a decimal),
- $ n $ is the number of compounding periods per year,
- $ t $ is the number of years.

We will calculate the compound amounts for both banks after 4 years.

Let's calculate these values.### Part A: Effective Annual Rates (EAR)
- **Bank A:** The effective annual rate (EAR) is **2.625%**.
- **Bank B:** The effective annual rate (EAR) is **2.624%**.

### Part B: Compound Amount After 4 Years
- **Bank A:** The compound amount after 4 years is **$5546.14**.
- **Bank B:** The compound amount after 4 years is **$5545.76**.

These results show that Bank A provides a slightly higher compound amount after 4 years due to its slightly higher effective annual rate.

Do you have any questions or would you like more details on any part of this solution?

Here are 5 related questions to further your understanding:
1. What is the impact of increasing the compounding frequency on the effective annual rate?
2. How would the compound amounts change if the investment period was 10 years instead of 4?
3. What happens to the compound amount if the nominal interest rate increases by 0.5% for both banks?
4. How does the effective annual rate differ between daily and continuous compounding?
5. What would the compound amount be for a different initial investment, such as $10,000?

**Tip:** Always compare the effective annual rate (EAR) rather than the nominal interest rate when evaluating investment options with different compounding frequencies.

Learn how to calculate effective rates and compound amounts for CDs with different compounding frequencies. This problem covers interest rates, compounding methods, and effective annual rate calculations for Bank A and Bank B. Suitable for students in Grades 10-12.

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